The invention relates to printing, particularly to micro-contact printing, also called xe2x80x9csoft lithographyxe2x80x9d, in which a flexible stamp transfers an xe2x80x9cinkedxe2x80x9d pattern to a receiving surface by mechanical contact, the pattern often having very small features normally associated with optical lithography and other expensive methods. More generally, the invention relates to a precise and controlled way of bringing two surfaces into contact, and subsequently separating them.
A number of printing techniques collectively known as xe2x80x9csoft lithographyxe2x80x9d have been recently developed, spurred by the 1993 discovery of micro-contact printing, as described in A. Kumar and G. M. Whitesides, FEATURES ON GOLD HAVING MICROMETER TO CENTIMETER DIMENSIONS CAN BE FORMED THROUGH A COMBINATION OF STAMPING WITH AN ELASTOMERIC STAMP AND AN ALKANETHIOL INK FOLLOWED BY CHEMICAL ETCHING, Appl. Phys. Lett., 63, 2002 (1993), the disclosure of which is incorporated by reference herein in their entirety. Typically, in such a printing technique, a flexible, polymeric stamp, embossed with a pattern and coated with a chemical xe2x80x9cinkxe2x80x9d, is brought into contact with a receiving surface and then separated from it, thereby transferring the image to the receiving surface in the form of a molecular monolayer of the ink. A full review of the techniques of soft lithography has recently been given in Y. Xia and G. M. Whitesides, SOFT LITHOGRAPHY, Angew. Chem. Int. Ed., 37, 550 (1998) and in B. Michel, et al., PRINTING MEETS LITHOGRAPHY: SOFT APPROACHES TO HIGH RESOLUTION PATTERNING, to be published in IBM Journal of Research and Development (special issue on lithography), the disclosures of both of which are incorporated by reference herein in their entirety.
Soft lithography promises to deliver printing that is less costly than that available with conventional techniques, such as optical lithography, used routinely in semiconductor processing. Soft lithography""s lower cost is possible because the per-print process is simpler than conventional techniquesxe2x80x94there are fewer steps and fewer costly machines. Moreover, soft lithography can print large areas quickly, whereas traditional, optical techniques can print only small areas at a time, and must build up large areas by xe2x80x9cstitchingxe2x80x9d (step and repeat), a slow process requiring an extremely precise and expensive machine known as a lithographic stepper.
To enable soft lithography, a printing method and apparatus are required to bring the stamp and the receiver into intimate contact, in a controlled and repeatable manner, such that the pattern on the stamp is transferred to the receiver with the greatest possible fidelity (i.e., with minimal distortion). To insure intimate contact everywhere, the printing method must prevent the trapping of gaseous bubbles (e.g., air bubbles) between the opposing surfaces of the stamp and the receiver. To insure repeatability, the printing apparatus must be automated. To achieve high fidelity, two requirements must be met. Firstly, the stamp itself should resist distortions in its own plane; such resistance is provided, for example, by the two-layer xe2x80x9chybrid stampsxe2x80x9d described by B. Michel et. al., supra. Secondly, the printing apparatus must provide, when the stamp and the receiver come into contact, uniform contact pressure and uniform geometric conditions over the entire printed area, lest the stamp be non-uniformly strained and therefore the printed pattern distorted.
Several prior-art methods and machines attempt to provide the printing requirements needed for soft lithography. However, these prior-art methods are deficient in several respects. One such method is described in U.S. Pat. No. 5,669,303 entitled APPARATUS AND METHOD FOR STAMPING A SURFACE, issued Sep. 23, 1997. This apparatus brings a circular stamp, held at its edges, into gradual contact with a receiver. The stamp is treated as a membrane under variable pressure: the convex (lower-pressure) side of the curving stamp being gradually flattened against the receiver while the periphery of the stamp is held fixed. Although the gradual contact successfully eliminates the trapping of air bubbles, this method and apparatus clearly produces non-uniform strain in the stamp as the varying pressure stretches the membrane, thereby distorting the pattern. Acknowledging this distortion, various schemes were proposed to compensate it, but the manufacturing practicality of these schemes is doubtful, and it would clearly be preferable if the method did not engender the non-uniform strain in the first place.
Another prior-art apparatus and method are described by B. Michel et al., supra, as the xe2x80x9crocker cylinder printing toolxe2x80x9d. In this method, the stamp is wrapped on a partial drum of radius R, and then xe2x80x9crockedxe2x80x9d upon the receiving surface in a manner somewhat analogous to the motion of a rocking chair upon a floor. In other words, the method is like a printing press in which the receiver remains stationary while the axis of the rotating drum translates over it. The problems with this method are three-fold. Firstly, the embossed pattern on the stamp is stretched in the print direction due to the drum""s curvature, introducing systematic distortion. Secondly, over the print cycle, the peak contact pressure between the stamp and the receiver is spatially non-uniform because it depends critically on the drum-to-receiver gap, which varies as the mechanism moves on account of unavoidable mechanical tolerances such as bearing runout and machining inaccuracies on the drum""s surface. Attempting to minimize variations in peak contact pressure by introducing a compliant layer (known as a xe2x80x9csoft padxe2x80x9d) behind the stamp simply trades peak-pressure non-uniformity for geometric non-uniformity; that is, as the soft pad compresses to accommodate gap changes, the local curvature of the stamp near the line of contact varies, and thus the tangential strain of the embossed pattern variesxe2x80x94this complex variation being superimposed on the systematic strain due to the drum""s curvature. Thirdly, because the drum is both translating and rotating, the accuracy of printing depends critically on precisely matching the drum""s translational speed xcexd with its rotational speed xcfx89; ideally, to roll without slipping and without straining the compliant stamp by frictional forces, the drum""s velocity xcexd should be exactly equal to xcfx89R. However, this ideal matching is nearly impossible to accomplish to the tolerance (xcx9c1 ppm) required for high-accuracy, large-area applicationsxe2x80x94exactly the applications where soft lithography seeks to replace optical lithogtaphy. Thus the rocker-style printer is ill-suited to the task of soft lithography. In fact, a controlled experiment was performed in which feature-placement errors on two prints from the same stamp were measuredxe2x80x94one print made with a well-engineered rocker printer, the other with an alternative scheme (such as the current invention), where the three problems discussed above are absent. The results demonstrate roughly a factor-of-three advantage in feature-placement accuracy for the latter method.
All three shortcomings of the rocker printer, of course, are shared by the xe2x80x9cprinting pressxe2x80x9d style of machine. In particular, the printing press shares the third shortcoming mentioned above (print accuracy dependent on precise matching of xcexd to xcfx89R): although the printing-press""s drum rotates without translating, the receiver instead translates beneath it, so speed matching is still an issue. Although the printing press is, of course, suitable for images to be observed by the human eye, where feature-placement accuracy need not be better than about 10 to 20 xcexcm, it appears to be unsuitable for the applications of soft lithography (e.g., printing patterns for electronic circuitry), where feature-placement accuracy on the order of 1 xcexcm or better is required.
Accordingly, there is a need for an improved method and apparatus for transferring patterns from a stamp to a receiver with great fidelity, the method and apparatus being scalable to large-size receivers and amenable of various types of stamps.
Disclosed is a printing apparatus, comprising a print surface lying in a print plane defined by an imaginary x-axis and y-axis, the print surface having an outward normal pointing in the positive direction along an imaginary z-axis, such that the x-axis, y-axis, and z-axis are substantially orthogonal to one another, a lower stamp clamp disposed adjacent to the negative-x edge of the print surface, an upper stamp clamp, moveable in two dimensions in a trajectory plane defined by the x-axis and z-axis, a stamp comprising a flexible material, the stamp having a first end attached to the lower stamp clamp and a second end attached to the upper stamp clamp, such that a cross section of the stamp parallel to the trajectory plane forms an arc extending from an origin point Q on the lower stamp clamp having (x,z) coordinates (0,0) to point E on the upper stamp clamp, this arc being described by the mathematical function xcex8(s), where s is the curvilinear distance along the arc measured from point Q, and xcex8 is the angle between the print plane and an imaginary line, the imaginary line being tangent to the cross section of the stamp at s, and wherein, during a print operation, the upper stamp clamp is moved in a trajectory comprising a plurality of xz positions of the upper clamp stamp that blend into a substantially continuous motion, the trajectory being effective in laying the stamp down smoothly and flat upon the print surface in a manner such that a moving contact front between the stamp and the print surface is created, the contact front being disposed substantially along a line characterized by a contact-front coordinate s0 x0 that increases as the trajectory progresses, the trajectory also being effective in causing the curvature       ⅆ    θ        ⅆ    s  
of the stamp at or near the contact front to be substantially constant throughout the motion.
Another aspect of the printer comprises a print-force-application system effective in pressing the stamp against the print surface, and defining an approximate contact front disposed substantially along a line lB parallel to the y-axis in the xy plane, the line lB intersecting the trajectory plane at (x, z)=(xB, 0), the approximate-contact-front x-coordinate xB increasing as the trajectory progresses and being substantially equal, at any stage of the trajectory, to the arc-length coordinate sB of point B, inasmuch as the arc of the stamp is assumed to be substantially flat over the segment from point Q to point B.
Another aspect of the printer further comprises a stamp-control system movable along the x-axis, wherein, throughout the trajectory, each xz position of the upper stamp clamp is a function of the displacement xC of the stamp-control system along the x-axis; the trajectory being effective in laying the stamp down upon the print surface such that the stamp is in continuous contact with a contact surface of the stamp-control system throughout the trajectory, the location of the contact surface being characterized by an arc-length coordinate sC that increases as the trajectory progresses.
In another aspect of the printer, the stamp-control system is disposed along a line lC parallel to the y-axis, line lC intersecting the trajectory plane at point C having coordinates xC and zC, where zC is a fixed, positive z-coordinate during any one printing operation, whereas xC increases as the trajectory progresses, in coordination with the contact-front coordinate x0.
In another aspect of the printer, the contact surface of the stamp-control system is a plane delimited in the x direction by two lines lC and lD separated by a fixed distance WCD, these lines being parallel to the y-axis and intersecting the trajectory plane at points C and D respectively, these points having coordinates (xC, zC) and (xD, zD) respectively, such that the contact surface is defined by the three parameters (xC, zC, xcex8CD), where       θ    CD    ≡            tan              -        1              ⁡          (                                    z            D                    -                      z            C                                                x            D                    -                      x            C                              )      
is the angle between the contact surface and the print plane, and such that the stamp angle xcex8(s) between arc-length coordinates s=sC and s=sD is substantially equal to xcex8CD; that is,
xcex8(s)≈xcex8CD for sCxe2x89xa6sxe2x89xa6sD. 
In another aspect of the printer, the upper stamp clamp is pivoted about a pivot line lP parallel to the y axis and intersecting the xz plane at point P having coordinates xP and zP; the stamp attaching to the upper stamp clamp along an upper-clamp line lE parallel to the y axis and intersecting the xz plane at point E having coordinates xE and zE; the upper-clamp line lE being disposed on the upper stamp clamp at a radius RS from the pivot line lP, such that the total arc length sE from the lower stamp clamp to the line lE is sE "Quadbond"L, where L is the known, free length of the stamp; and wherein the stamp attaches to the upper-clamp line lE at an angle xcex8Exe2x89xa1xcex8(L).
In another aspect of the printer, the trajectory comprises a plurality of configurations, each configuration described by the coordinate s0"Quadbond"x0 of the contact front and by corresponding coordinates xP, zP of the pivot line given by the equations
xP=xE+Rs cos xcex8E 
zP=zE+Rs sin xcex8E, 
where             x      E        =                            ∫          0          L                ⁢                  cos          ⁢                      xe2x80x83                    ⁢                      θ            ⁡                          (              s              )                                ⁢                      ⅆ            s                    ⁢                      xe2x80x83                    ⁢          and          ⁢                      xe2x80x83                    ⁢                      z            E                              =                        ∫          0          L                ⁢                  sin          ⁢                      xe2x80x83                    ⁢                      θ            ⁡                          (              s              )                                ⁢                      ⅆ            s                                ,
and where the mathematical function xcex8(s) describing the shape of the arc for a given configuration is assumed to be
xcex8(s)=0 for 0xe2x89xa6sxe2x89xa6s0, 
whereas for s greater than s0, xcex8(s) is determined by solution of the differential equations                     ⅆ        u                    ⅆ        s              =          F      ⁡              (        u        )              ,
the lower-end boundary conditions                     u        0            ≡              {                                                            u                10                                                                                        u                20                                                    }            ≡              {                                                            θ                0                                                                                                                              ⅆ                    θ                                                        ⅆ                    s                                                  ⁢                                  |                  0                                                                    }              =          {                                    0                                                              κ              0                                          }        ,
and the upper-end boundary condition                                           T            ⁡                          (              β              )                                ≡                      E            ⁢                          xe2x80x83                        ⁢            I            ⁢                                          ⅆ                θ                                            ⅆ                s                                                    ⁢                  |          E                ⁢                                            +                              F                X0                                      ⁢                          R              S                        ⁢            sin            ⁢                          xe2x80x83                        ⁢                          θ              E                                -                                    w              ⁡                              (                                  s                  -                                      s                    0                                                  )                                      ⁢                          R              s                        ⁢            cos            ⁢                          xe2x80x83                        ⁢                          θ              E                                          =      0        ,          
        ⁢    wherein              u      ≡              {                                                            u                1                                                                                        u                2                                                    }            ≡              {                                            θ                                                                                            ⅆ                  θ                                                  ⅆ                  s                                                                    }              ,                  F        ⁡                  (          u          )                    ≡              {                                                            u                2                                                                                                                                                    F                      X0                                                              E                      ⁢                                              xe2x80x83                                            ⁢                      I                                                        ⁢                  sin                  ⁢                                      xe2x80x83                                    ⁢                                      u                    1                                                  -                                                                            w                      ⁡                                              (                                                  s                          -                                                      s                            0                                                                          )                                                                                    E                      ⁢                                              xe2x80x83                                            ⁢                      I                                                        ⁢                  cos                  ⁢                                      xe2x80x83                                    ⁢                                      u                    1                                                                                      }              ,  
xcexa0 is a specified curvature at point O, the parameter xcex2xe2x89xa1FX0, unknown a priori, is the internal x-directed force acting on the stamp""s cross section at s=s0 per unit depth of the stamp in the y direction, E is Young""s modulus of the stamp, I is the area moment of inertia of the stamp""s cross section per unit depth in the y-direction, and w is the weight per unit area of the stamp; and
wherein for each configuration the solution for xP and zP is derived by means of the xe2x80x9cshooting methodxe2x80x9d, whereby an initial value xcex2(0) of xcex2 is guessed, the differential equations are solved to yield T(xcex2(0)) and             [                        ∂          T                          ∂          β                    ]              β      =              β                  (          0          )                            -      1        ,
Newton iteration       β          (              n        +        1            )        =            β              (        n        )              -                            [                                    ∂              T                                      ∂              β                                ]                          β          =                      β                          (              n              )                                                -          1                    ⁢              T        ⁡                  (                      β                          (              n              )                                )                    
is applied to obtain a refined value xcex2(1) of the unknown parameter xcex2, whereupon the differential equations are solved again; this iteration procedure being applied repeatedly until the correct auxiliary boundary condition T(xcex2)=0 is achieved to within some tolerance.
In another aspect of the printer the upper stamp clamp is pivoted about a pivot line lP parallel to the y axis and intersecting the xz plane at point P having coordinates xP and zP; the stamp attaching to the upper stamp clamp along an upper-clamp line lE parallel to the y axis and intersecting the xz plane at point E having coordinates xE and zE; the upper-clamp line lE being disposed on the upper stamp clamp at a radius RS from the pivot line lP, such that the total arc length sE from the lower stamp clamp to the line lE is sE"Quadbond"L, where L is the known, free length of the stamp; and wherein the stamp attaches to the upper-clamp line lE at an angle xcex8xe2x89xa1xcex8(L).
In another aspect of the printer, the trajectory comprises a plurality of configurations, each configuration described by the coordinate sB"Quadbond"xB of the approximate contact front and by corresponding coordinates xP, zP of the pivot line given by the equations
xP=xE+Rs cos xcex8E 
zP=zE+Rs sin xcex8E, 
where             x      E        =                            ∫          0          L                ⁢                  cos          ⁢                      xe2x80x83                    ⁢                      θ            ⁡                          (              s              )                                ⁢                      ⅆ            s                    ⁢                      xe2x80x83                    ⁢          and          ⁢                      xe2x80x83                    ⁢                      z            E                              =                        ∫          0          L                ⁢                  sin          ⁢                      xe2x80x83                    ⁢                      θ            ⁡                          (              s              )                                ⁢                      ⅆ            s                                ,
and where the mathematical function xcex8(s)
describing the shape of the arc for a given configuration is assumed to be
xcex8(s)=0 for 0xe2x89xa6sxe2x89xa6sB, 
whereas for s greater than sB, xcex8(s) is determined by solution of the differential equations                     ⅆ        u                    ⅆ        s              =          F      ⁡              (        u        )              ,
the lower-end boundary conditions                     u        B            ≡              {                                                            u                                  1                  ⁢                  B                                                                                                        u                                  2                  ⁢                  B                                                                    }            ≡              {                                                            θ                B                                                                                                                              ⅆ                    θ                                                        ⅆ                    s                                                  ⁢                                  |                  B                                                                    }              =          {                                    0                                                              κ              B                                          }        ,
and the upper-end boundary condition                                           T            ⁡                          (              β              )                                ≡                      E            ⁢                          xe2x80x83                        ⁢            I            ⁢                                          ⅆ                θ                                            ⅆ                s                                                    ⁢                  |          E                ⁢                                            +                              F                XB                                      ⁢                          R              S                        ⁢            sin            ⁢                          xe2x80x83                        ⁢                          θ              E                                -                                    w              ⁡                              (                                  s                  -                                      s                    B                                                  )                                      ⁢                          R              s                        ⁢            cos            ⁢                          xe2x80x83                        ⁢                          θ              E                                          =      0        ,          
        ⁢    wherein              u      ≡              {                                                            u                1                                                                                        u                2                                                    }            ≡              {                                            θ                                                                                            ⅆ                  θ                                                  ⅆ                  s                                                                    }              ,                  F        ⁡                  (          u          )                    ≡              {                                                            u                2                                                                                                                                                    F                      XB                                                              E                      ⁢                                              xe2x80x83                                            ⁢                      I                                                        ⁢                  sin                  ⁢                                      xe2x80x83                                    ⁢                                      u                    1                                                  -                                                                            w                      ⁡                                              (                                                  s                          -                                                      s                            B                                                                          )                                                                                    E                      ⁢                                              xe2x80x83                                            ⁢                      I                                                        ⁢                  cos                  ⁢                                      xe2x80x83                                    ⁢                                      u                    1                                                                                      }              ,  
xcexaB is a specified curvature at point B, the parameter xcex2xe2x89xa1FXB, unknown a priori, is the internal x-directed force acting on the stamp""s cross section at s=sB per unit depth of the stamp in the y direction, E is Young""s modulus of the stamp, I is the area moment of inertia of the stamp""s cross section per unit depth in the y-direction, and w is the weight per unit area of the stamp; and
wherein for each configuration the solution for xP and zP is derived by means of the xe2x80x9cshooting methodxe2x80x9d, whereby an initial value xcex2(0) of xcex2 is guessed, the differential equations are solved to yield T (xcex2(0)) and             [                        ∂          T                          ∂          β                    ]              β      =              β                  (          0          )                            -      1        ,
Newton iteration       β          (              n        +        1            )        =            β              (        n        )              -                            [                                    ∂              T                                      ∂              β                                ]                          β          =                      β                          (              n              )                                                -          1                    ⁢              T        ⁡                  (                      β                          (              n              )                                )                    
is applied to obtain a refined value xcex2(1) of the unknown parameter xcex2, whereupon the differential equations are solved again; this iteration procedure being applied repeatedly until the correct auxiliary boundary condition T(xcex2)=0 is achieved to within some tolerance.
In another aspect of the printer, the upper stamp clamp is pivoted about a pivot line lP parallel to the y axis and intersecting the xz plane at point P having coordinates xP and zP; the stamp attaching to the upper stamp clamp along an upper-clamp line lE parallel to the y axis and intersecting the xz plane at point E having coordinates xE and zE, the upper-clamp line lE being disposed on the upper stamp clamp at a radius RS from the pivot line lP, such that the total arc length sE from the lower stamp clamp to the line lE is sE"Quadbond"L, where L is the known, free length of the stamp; and wherein the stamp attaches to the upper-clamp line lE at an angle xcex8Exe2x89xa1xcex8(L).
In another aspect of the printer, the trajectory comprises a plurality of configurations, each configuration described by the coordinate s0"Quadbond"x0 of the contact front and by corresponding coordinates xP, zP of the pivot line given by the equations:
xP=xE+Rs cos xcex8E 
zP=zE+Rs sin xcex8E, 
where             x      E        =                            ∫          0          L                ⁢                  cos          ⁢                      xe2x80x83                    ⁢                      θ            ⁡                          (              s              )                                ⁢                      ⅆ            s                    ⁢                      xe2x80x83                    ⁢          and          ⁢                      xe2x80x83                    ⁢                      z            E                              =                        ∫          0          L                ⁢                  sin          ⁢                      xe2x80x83                    ⁢                      θ            ⁡                          (              s              )                                ⁢                      ⅆ            s                                ,
and where the mathematical function xcex8(s) describing the shape of the arc for a given configuration is assumed to be
xcex8(s)=0 for 0xe2x89xa6sxe2x89xa6s0, 
whereas for s greater than s0, xcex8(s) is determined by solution of the differential equations                     ⅆ        u                    ⅆ        s              =          F      ⁡              (        u        )              ,
the lower-end boundary conditions                     u        0            ≡              {                                                            u                10                                                                                        u                20                                                                                        u                30                                                                                        u                40                                                    }            ≡              {                                                            θ                0                                                                                                                              ⅆ                    θ                                                        ⅆ                    s                                                  ⁢                                  |                  0                                                                                                        F                                  X                  0                                                                                                        F                                  Z                  0                                                                    }              =          {                                    0                                                0                                                              F                              X                0                                                                                        F                              Z                0                                                        }        ,
and the auxiliary boundary conditions
T(xcex2)=0, 
wherein       u    ≡          {                                                  u              1                                                                          u              2                                                                          u              3                                                                          u              4                                          }        ≡          {                                                  θ              0                                                                                          ⅆ                θ                                            ⅆ                s                                                                                                        F                x                            ⁡                              (                s                )                                                                                                        F                z                            ⁡                              (                s                )                                                        }        ,            F      ⁡              (        u        )              ≡          {                                                  u              2                                                                                                                                u                    3                                                        E                    ⁢                                          xe2x80x83                                        ⁢                    I                                                  ⁢                sin                ⁢                                  xe2x80x83                                ⁢                                  u                  1                                            -                                                                    u                    4                                                        E                    ⁢                                          xe2x80x83                                        ⁢                    I                                                  ⁢                cos                ⁢                                  xe2x80x83                                ⁢                                  u                  1                                                                                                                                          -                                      p                    ⁡                                          (                      s                      )                                                                      ⁢                sin                ⁢                                  xe2x80x83                                ⁢                                  u                  1                                            -                                                f                  ⁡                                      (                    s                    )                                                  ⁢                                  xe2x80x83                                ⁢                cos                ⁢                                  xe2x80x83                                ⁢                                  u                  1                                                                                                        w              +                                                p                  ⁡                                      (                    s                    )                                                  ⁢                                  xe2x80x83                                ⁢                cos                ⁢                                  xe2x80x83                                ⁢                                  u                  1                                            -                                                f                  ⁡                                      (                    s                    )                                                  ⁢                                  xe2x80x83                                ⁢                sin                ⁢                                  xe2x80x83                                ⁢                                  u                  1                                                                        }        ,            T      ⁡              (        β        )              ≡          {                                                                  z                C                            -                                                ∫                  0                                      S                    C                                                  ⁢                                  sin                  ⁢                                      xe2x80x83                                    ⁢                                      θ                    ⁡                                          (                      s                      )                                                        ⁢                                      xe2x80x83                                    ⁢                                      ⅆ                    s                                                                                                                                          θ                C                            -                              θ                                  C                  ⁢                                      xe2x80x83                                    ⁢                  D                                                                                                                        E                ⁢                                  xe2x80x83                                ⁢                I                ⁢                                                      ⅆ                    θ                                                        ⅆ                    s                                                              ⁢                              |                E                            ⁢                                                                    +                                          F                                              X                        ⁢                                                  xe2x80x83                                                ⁢                        E                                                                              ⁢                                      R                    S                                    ⁢                  sin                  ⁢                                      xe2x80x83                                    ⁢                                      θ                    E                                                  -                                                      F                    ZE                                    ⁢                                      R                    S                                    ⁢                  cos                  ⁢                                      xe2x80x83                                    ⁢                                      θ                    E                                                                                          }        ,
and wherein FX(s) and FZ(s) are functions of s describing the internal x-directed and z-directed forces acting on the stamp""s cross section at s per unit depth of the stamp in the y direction, FXExe2x89xa1FX(sE), FZE(sE), xcex2 is a vector of parameters that are unknown a priori,       β    =          {                                                  s              0                                                                          F              X0                                                                          F              Z0                                          }        ,
s0 is the aforementioned arc-length coordinate of the contact front, FX0xe2x89xa1FX(s0), FZ0xe2x89xa1FZ(s0), E is Young""s modulus of the stamp, I is the area moment of inertia of the stamp""s cross section per unit depth in the y-direction, w is the weight per unit area of the stamp, p(s) and f(s) are functions of s describing forces applied normal to the stamp and tangential to the stamp respectively by the print-force-application system, the stamp-control system and the print surface, sC is the value of arc-length coordinate a at point C, xcex8Cxe2x89xa1xcex8(sC) is the angle of the arc at point C, and xcex8CD is the aforementioned angle of the stamp-control system""s contact surface; and
wherein for each configuration the solution for xP and zP is derived by means of the xe2x80x9cshooting methodxe2x80x9d, whereby an initial value xcex2(0) of xcex2 is guessed, the differential equations are solved to yield T(xcex2(0)) and             [                        ∂          T                          ∂          β                    ]              β      =              β                  (          0          )                            -      1        ,
Newton-Raphson iteration       β          (              n        +        1            )        =            β              (        n        )              -                            [                                    ∂              T                                      ∂              β                                ]                          β          =                      β                          (              n              )                                                -          1                    ⁢              T        ⁡                  (                      β                          (              n              )                                )                    
is applied to obtain a refined vector xcex2(1) whereupon the differential equations are solved again; this iteration procedure being applied repeatedly until the correct auxiliary boundary conditions T(xcex2)=0 are achieved to within some tolerance.
In another aspect of the printer, the trajectory comprises a plurality of configurations, each configuration described by the coordinate s0"Quadbond"x0 of the contact front and by corresponding coordinates xP, zP of the pivot line given by the equations:
xP=xE+Rs cos xcex8E 
zP=zE+Rs sin xcex8E, 
where             x      E        =                            ∫          0          L                ⁢                  cos          ⁢                      xe2x80x83                    ⁢                      θ            ⁡                          (              s              )                                ⁢                      ⅆ            s                    ⁢                      xe2x80x83                    ⁢          and          ⁢                      xe2x80x83                    ⁢                      z            E                              =                        ∫          0          L                ⁢                  sin          ⁢                      xe2x80x83                    ⁢                      θ            ⁡                          (              s              )                                ⁢                      ⅆ            s                                ,
and where the mathematical function xcex8(s) describing the shape of the arc for a given configuration is assumed to be
xcex8(s)=0 for 0xe2x89xa6sxe2x89xa6s0, 
whereas for s greater than s0, xcex8(s) is determined in stamp segments OC and DE by solution of the differential equations                     ⅆ        u                    ⅆ        s              =          F      ⁡              (        u        )              ,
the lower-end boundary conditions                     u        0            ≡              {                                                            u                10                                                                                        u                20                                                    }            ≡              {                                                            θ                0                                                                                                                              ⅆ                    θ                                                        ⅆ                    s                                                  ⁢                                  |                  0                                                                    }              =          {                                    0                                                              κ              0                                          }        ,
and the upper-end boundary condition                                           T            ⁡                          (              β              )                                ≡                      EI            ⁢                                          ⅆ                θ                                            ⅆ                s                                                    ⁢                  |          E                ⁢                                            +                              F                XE                                      ⁢                          R              S                        ⁢            sin            ⁢                          xe2x80x83                        ⁢                          θ              E                                -                                    F              ZE                        ⁢                          R              S                        ⁢            cos            ⁢                          xe2x80x83                        ⁢                          θ              E                                          =      0        ,          
        ⁢    wherein              u      ≡              {                                                            u                1                                                                                        u                2                                                    }            ≡              {                                            θ                                                                                            ⅆ                  θ                                                  ⅆ                  s                                                                    }              ,          
        ⁢                  F        ⁡                  (          u          )                    ≡              {                                                                              u                  2                                                                                                                                                                                                  F                          X                                                ⁡                                                  (                          s                          )                                                                    EI                                        ⁢                    sin                    ⁢                                          xe2x80x83                                        ⁢                                          u                      1                                                        -                                                                                                              F                          Z                                                ⁡                                                  (                          s                          )                                                                    EI                                        ⁢                    cos                    ⁢                                          xe2x80x83                                        ⁢                    u                                                                                1                }              ,  
xcexa0 is a specified curvature at point O, E is Young""s modulus of the stamp, I is the area moment of inertia of the stamp""s cross section per unit depth in the y-direction, w is the weight per unit area of the stamp, Fx(s) and Fz(s) are the x-directed and z-directed stamp forces per unit length of stamp in the y direction, given by             F      x        ⁡          (      s      )        =      {                                                                                      ⁢                                                      F                    x0                                    ,                                                                                    0                ≤                s                ≤                                  s                  C                                                                                                                        ⁢                                                                            F                      x0                                        +                                          Δ                      ⁢                                              xe2x80x83                                            ⁢                                              F                        x                                                                              ,                                                                                                                          s                    D                                    ≤                  s                  ≤                                      s                    E                                                  ,                                                    ⁢                  
                ⁢        and        ⁢                  
                ⁢                              F            z                    ⁡                      (            s            )                              =              {                                                                            ⁢                                  0                  ,                                                                                    0                ≤                s                ≤                                  s                  0                                                                                                                        ⁢                                                      w                    ⁡                                          (                                              s                        -                                                  s                          0                                                                    )                                                        ,                                                                                                      s                  0                                ≤                s                ≤                                  s                  C                                                                                                                        ⁢                                                                            w                      ⁡                                              (                                                  s                          -                                                      s                            0                                                                          )                                                              +                                          Δ                      ⁢                                              xe2x80x83                                            ⁢                                              F                        z                                                                              ,                                                                                                                          s                    D                                    ≤                  s                  ≤                                      s                    E                                                  ,                                                        
in which Fz0xe2x89xa1Fx(s0)xe2x89xa1xcex2 is a parameter that is unknown a priori, and the differences xcex94Fx and xcex94Fz are respectively the differences
xcex94Fxxe2x89xa1Fx(sD)xe2x88x92Fx(sC) 
xcex94Fzxe2x89xa1Fz(sD)xe2x88x92Fz(sC) 
that occur across stamp segment CD where the stamp-control system contacts the stamp, the values of which differences, along with the value of the difference                     Δ        ⁢                  xe2x80x83                ⁢        κ            ≡                        ⅆ          θ                          ⅆ          s                      ⁢          |      D        ⁢          -                        ⅆ          θ                          ⅆ          s                      ⁢          |      C        ,
may be calculated from the three equations of static equilibrium for the stamp under the action of forces applied to the stamp by the stamp-control system, these three differences together with xcex8D=xcex8C permitting numerical integration for stamp segment DE to proceed immediately from the numerical-integration result obtained at the final point C in stamp segment OC, and wherein for each configuration the solution for xP and zP is derived by means of the xe2x80x9cshooting methodxe2x80x9d, whereby an initial value xcex2(0) of xcex2 is guessed, the differential equations are solved to yield T(xcex2(0)) and             [                        ∂          T                          ∂          β                    ]              β      =              β                  (          0          )                            -      1        ,
Newton iteration       β          (              n        +        1            )        =            β              (        n        )              -                            [                                    ∂              T                                      ∂              β                                ]                          β          =                      β                          (              n              )                                                -          1                    ⁢              T        ⁡                  (                      β                          (              n              )                                )                    
is applied to obtain a refined vector xcex2(1), whereupon the differential equations are solved again; this iteration procedure being applied repeatedly until the correct auxiliary boundary conditions T(xcex2)=0 are achieved to within some tolerance.
In another aspect of the printer, a stamp-control system movable along the x-axis, wherein, throughout the trajectory, each xz position of the upper stamp clamp is a function of the displacement xC of the stamp-control system along the x-axis; the trajectory being effective in laying the stamp down upon the print surface such that the stamp is in continuous contact with a contact surface of the stamp-control system, the location of the contact surface being characterized by an arc-length coordinate sC that increases as the trajectory progresses.
In another aspect of the printer, the stamp-control system is disposed along a line lC parallel to the y-axis, line lC intersecting the trajectory plane at point C having coordinates xC and zC, where zC is a fixed, positive z-coordinate during any one printing operation, whereas xC increases as the trajectory progresses, in coordination with the contact-front coordinate x0.
In another aspect of the printer the contact surface of the stamp-control system is a plane delimited in the x direction by two lines lC and lD separated by a fixed distance WCD, these lines being parallel to the y-axis and intersecting the trajectory plane at points C and D respectively, these points having coordinates (xC, zC) and (xD, zD) respectively, such that the contact surface is defined by the three parameters (xC, zC, xcex8CD), where       θ    CD    ≡            tan              -        1              ⁡          (                                    z            D                    -                      z            C                                                x            D                    -                      x            C                              )      
is the angle between the contact surface and the print plane, and such that the stamp angle xcex8(s) between arc-length coordinates s=sC and s=sD is substantially equal to xcex8CD; that is,
xcex8(s)≈xcex8CD for sCxe2x89xa6sxe2x89xa6sD. 
In another aspect of the printer, the upper stamp clamp is pivoted about a pivot line lP parallel to the y axis and intersecting the xz plane at point P having coordinates xP and zP; the stamp attaching to the upper stamp clamp along an upper-clamp line lE parallel to the y axis and intersecting the xz plane at point E having coordinates xE and zE; the upper-clamp line lE being disposed on the upper stamp clamp at a radius RS from the pivot line lP, such that the total arc length sE from the lower stamp clamp to the line lE is sE"Quadbond"L, where L is the known, free length of the stamp; and wherein the stamp attaches to the upper-clamp line lE at an angle xcex8Exe2x89xa1xcex8(L).
In another aspect of the printer, the trajectory comprises a plurality of configurations, each configuration described by the coordinate s0"Quadbond"x0 of the contact front and by corresponding coordinates xP, zP of the pivot line given by the equations:
xP=xE+Rs cos xcex8E 
zP=zE+Rs sin xcex8E, 
where             x      E        =                            ∫          0          L                ⁢                  cos          ⁢                      xe2x80x83                    ⁢                      θ            ⁡                          (              s              )                                ⁢                      ⅆ            s                    ⁢                      xe2x80x83                    ⁢          and          ⁢                      xe2x80x83                    ⁢                      z            E                              =                        ∫          0          L                ⁢                  sin          ⁢                      xe2x80x83                    ⁢                      θ            ⁡                          (              s              )                                ⁢                      ⅆ            s                                ,
and where the mathematical function xcex8(s) describing the shape of the arc for a given configuration is assumed to be
xcex8(s)=0 for 0xe2x89xa6sxe2x89xa6s0, 
whereas for s greater than s0, xcex8(s) is determined by solution of the differential equations                     ⅆ        u                    ⅆ        s              =          F      ⁡              (        u        )              ,
the lower-end boundary conditions                     u        0            ≡              {                                                            u                10                                                                                        u                20                                                                                        u                30                                                                                        u                40                                                    }            ≡              {                                                            θ                0                                                                                                                              ⅆ                    θ                                                        ⅆ                    s                                                  ⁢                                  |                  0                                                                                                        F                                  X                  0                                                                                                        F                                  Z                  0                                                                    }              =          {                                    0                                                0                                                              F                              X                0                                                                                        F                              Z                0                                                        }        ,
and the auxiliary boundary conditions
T(xcex2)=0, 
wherein       u    ≡          {                                                  u              1                                                                          u              2                                                                          u              3                                                                          u              4                                          }        ≡          {                                    θ                                                                              ⅆ                θ                                            ⅆ                s                                                                                                        F                x                            ⁡                              (                s                )                                                                                                        F                z                            ⁡                              (                s                )                                                        }        ,            F      ⁡              (        u        )              ≡          {                                                  u              2                                                                                                                                u                    3                                    EI                                ⁢                sin                ⁢                                  xe2x80x83                                ⁢                                  u                  1                                            -                                                                    u                    4                                    EI                                ⁢                cos                ⁢                                  xe2x80x83                                ⁢                                  u                  1                                                                                                                                          -                                      p                    ⁡                                          (                      s                      )                                                                      ⁢                sin                ⁢                                  xe2x80x83                                ⁢                                  u                  1                                            -                                                f                  ⁡                                      (                    s                    )                                                  ⁢                cos                ⁢                                  xe2x80x83                                ⁢                                  u                  1                                                                                                        w              +                                                p                  ⁡                                      (                    s                    )                                                  ⁢                cos                ⁢                                  xe2x80x83                                ⁢                                  u                  1                                            -                                                f                  ⁡                                      (                    s                    )                                                  ⁢                sin                ⁢                                  xe2x80x83                                ⁢                                  u                  1                                                                        }        ,      
    ⁢            T      ⁡              (        β        )              ≡          {                                                                  z                C                            -                                                ∫                  0                                      s                    C                                                  ⁢                                  sin                  ⁢                                      xe2x80x83                                    ⁢                                      θ                    ⁡                                          (                      s                      )                                                        ⁢                                      ⅆ                    s                                                                                                                                          θ                C                            -                              θ                CD                                                                                                        EI                ⁢                                                      ⅆ                    θ                                                        ⅆ                    s                                                              ⁢                              |                E                            ⁢                                                                    +                                          F                      XE                                                        ⁢                                      R                    S                                    ⁢                  sin                  ⁢                                      xe2x80x83                                    ⁢                                      θ                    E                                                  -                                                      F                    ZE                                    ⁢                                      R                    S                                    ⁢                  cos                  ⁢                                      xe2x80x83                                    ⁢                                      θ                    E                                                                                          }        ,
and wherein FX(s) and FZ(s) are functions of s describing the internal x-directed and z-directed forces acting on the stamp""s cross section at s per unit depth of the stamp in the y direction, FXExe2x89xa1FX(sE), FZExe2x89xa1FZ(sE), xcex2 is a vector of parameters that are unknown a priori,       β    =          {                                                  s              0                                                                          F                              X                0                                                                                        F                              Z                0                                                        }        ,
s0 is the aforementioned arc-length coordinate of the contact front, FX0xe2x89xa1FX(0), FZ0xe2x89xa1FZ(0), E is Young""s modulus of the stamp, I is the area moment of inertia of the stamp""s cross section per unit depth in the y-direction, w is the weight per unit area of the stamp, p(s) and f(s) are functions of s describing forces applied normal to the stamp and tangential to the stamp respectively by the print-force-application system, the stamp-control system and the print surface, sC is the value of arc-length coordinate s at point C, xcex8Cxe2x89xa1xcex8(sC) is the angle of the arc at point C, and xcex8CD is the aforementioned angle of the stamp-control system""s contact surface; and
wherein for each configuration the solution for xP and zP is derived by means of the xe2x80x9cshooting methodxe2x80x9d, whereby an initial value xcex2(0) of xcex2 is guessed, the differential equations are solved to yield T(xcex2(0)) and             [                        ∂          T                          ∂          β                    ]              β      =              β                  (          0          )                            -      1        ,
Newton-Raphson iteration       β          (              n        +        1            )        =            β              (        n        )              -                            [                                    ∂              T                                      ∂              β                                ]                          β          =                      β                          (              n              )                                                -          1                    ⁢              T        ⁡                  (                      β                          (              n              )                                )                    
is applied to obtain a refined vector xcex2(1), whereupon the differential equations are solved again; this iteration procedure being applied repeatedly until the correct auxiliary boundary conditions T(xcex2)=0 are achieved to within some tolerance.
In another aspect of the printer, the trajectory comprises a plurality of configurations, each configuration described by the coordinate sB"Quadbond"xB of the approximate contact front and by corresponding coordinates xP, zP of the pivot line given by the equations
xP=xE+Rs cos xcex8E 
zP=zE+Rs sin xcex8E, 
where             x      E        =                            ∫          0          L                ⁢                  cos          ⁢                      xe2x80x83                    ⁢                      θ            ⁡                          (              s              )                                ⁢                      ⅆ            s                    ⁢                      xe2x80x83                    ⁢          and          ⁢                      xe2x80x83                    ⁢                      z            E                              =                        ∫          0          L                ⁢                  sin          ⁢                      xe2x80x83                    ⁢                      θ            ⁡                          (              s              )                                ⁢                      ⅆ            s                                ,
and where the mathematical function xcex8(s) describing the shape of the arc for a given configuration is assumed to be
xcex8(s)=0 for 0xe2x89xa6sxe2x89xa6sB, 
whereas for s greater than sB, xcex8(s) is determined in stamp segments OC and DE by solution of the differential equations                     ⅆ        u                    ⅆ        s              =          F      ⁡              (        u        )              ,
the lower-end boundary conditions                     u        B            ≡              {                                                            u                                  1                  ⁢                  B                                                                                                        u                                  2                  ⁢                  B                                                                    }            ≡              {                                                            θ                B                                                                                                                              ⅆ                    θ                                                        ⅆ                    s                                                  ⁢                                  |                  B                                                                    }              =          {                                    0                                                              κ              B                                          }        ,
and the upper-end boundary condition                                           T            ⁡                          (              β              )                                ≡                      EI            ⁢                                          ⅆ                θ                                            ⅆ                s                                                    ⁢                  |          E                ⁢                                            +                              F                XE                                      ⁢                          R              S                        ⁢            sin            ⁢                          xe2x80x83                        ⁢                          θ              E                                -                                    F              ZE                        ⁢                          R              S                        ⁢            cos            ⁢                          xe2x80x83                        ⁢                          θ              E                                          =      0        ,          
        ⁢    wherein              u      ≡              {                                                            u                1                                                                                        u                2                                                    }            ≡              {                                            θ                                                                                            ⅆ                  θ                                                  ⅆ                  s                                                                    }              ,          
        ⁢                  F        ⁡                  (          u          )                    ≡              {                                                            u                2                                                                                                                                                                            F                        x                                            ⁡                                              (                        s                        )                                                              EI                                    ⁢                  sin                  ⁢                                      xe2x80x83                                    ⁢                                      u                    1                                                  -                                                                                                    F                        z                                            ⁡                                              (                        s                        )                                                              EI                                    ⁢                  cos                  ⁢                                      xe2x80x83                                    ⁢                                      u                    1                                                                                      }              ,  
xcexaB is a specified curvature at point B, E is Young""s modulus of the stamp, I is the area moment of inertia of the stamp""s cross section per unit depth in the y-direction, w is the weight per unit area of the stamp, Fx(s) and Fz(s) are the x-directed and z-directed stamp forces per unit length of stamp in the y direction, given by             F      x        ⁡          (      s      )        =      {                                                                                      ⁢                                                      F                    x0                                    ,                                                                                    0                ≤                s                ≤                                  s                  C                                                                                                                        ⁢                                                                            F                      x0                                        +                                          Δ                      ⁢                                              xe2x80x83                                            ⁢                                              F                        x                                                                              ,                                                                                                                          s                    D                                    ≤                  s                  ≤                                      s                    E                                                  ,                                                    ⁢                  
                ⁢        and        ⁢                  
                ⁢                              F            z                    ⁡                      (            s            )                              =              {                                                                            ⁢                                  0                  ,                                                                                    0                ≤                s                ≤                                  s                  0                                                                                                                        ⁢                                                      w                    ⁡                                          (                                              s                        -                                                  s                          0                                                                    )                                                        ,                                                                                                      s                  0                                ≤                s                ≤                                  s                  C                                                                                                                        ⁢                                                                            w                      ⁡                                              (                                                  s                          -                                                      s                            0                                                                          )                                                              +                                          Δ                      ⁢                                              xe2x80x83                                            ⁢                                              F                        z                                                                              ,                                                                                                                          s                    D                                    ≤                  s                  ≤                                      s                    E                                                  ,                                                        
in which Fx0xe2x89xa1Fx(s0)xe2x89xa1xcex2 is a parameter that is unknown a priori, and the differences xcex94Fx and xcex94Fz are respectively the differences
xcex94Fxxe2x89xa1Fx(sD)xe2x88x92Fx(sC) 
xcex94Fzxe2x89xa1Fz(sD)xe2x88x92Fz(sC) 
that occur across stamp segment CD where the stamp-control system contacts the stamp, the values of which differences, along with the value of the difference                     Δ        ⁢                  xe2x80x83                ⁢        κ            ≡                        ⅆ          θ                          ⅆ          s                      ⁢          |      D        ⁢          -                        ⅆ          θ                          ⅆ          s                      ⁢          |      C        ,
may be calculated from the three equations of static equilibrium for the stamp under the action of forces applied to the stamp by the stamp-control system, these three differences together with xcex8D=xcex8C permitting numerical integration for stamp segment DE to proceed immediately from the numerical-integration result obtained at the final point C in stamp segment OC, and wherein for each configuration the solution for xP and zP is derived by means of the xe2x80x9cshooting methodxe2x80x9d, whereby an initial value xcex2(0) of xcex2 is guessed, the differential equations are solved to yield T(xcex2(0)) and             [                        ∂          T                          ∂          β                    ]              β      =              β                  (          0          )                            -      1        ,
Newton iteration       β          (              n        +        1            )        =            β              (        n        )              -                            [                                    ∂              T                                      ∂              β                                ]                          β          =                      β                          (              n              )                                                -          1                    ⁢              T        ⁡                  (                      β                          (              n              )                                )                    
is applied to obtain a refined vector xcex2(1) whereupon the differential equations are solved again; this iteration procedure being applied repeatedly until the correct auxiliary boundary conditions T(xcex2)=0 are achieved to within some tolerance.
In another aspect of the printer, the print-force-application system comprises a flat-iron.
In another aspect of the printer, the stamp-control system comprises a vacuum bar.
Disclosed is a printing apparatus, comprising a receiver means whose receiving surface lies in an xy plane, the normal to the surface defining a z-axis direction, a lower stamp clamp means for fixing a first edge of a stamp, an upper stamp clamp means for holding a second edge of a stamp for movement in the xz directions, a flexible stamp means for printing to the receiver, said flexible stamp in substantially the form of a sheet defining edges, the first edge of which is affixed to the lower stamp clamp, and the opposing second edge of which is affixed to the upper stamp clamp, thereby allowing the stamp to hang in a curve under gravity and the sheet""s own stiffness, such that every normal to the stamp""s curved surface lies substantially parallel to the xz plane, and a trajectory-producing means for moving the upper stamp clamp along a prescribed trajectory in the xz plane, such that the stamp is draped upon the receiving surface in a manner that causes the curvature of the stamp near a contact front at a point B to be constant throughout the trajectory.
Another aspect of the apparatus further comprises print-force application means for applying pressure upon the stamp means against the receiver means and for defining the contact front.
Another aspect of the apparatus further comprises stamp-control means for defining a point C through which the curvature of the sheet will pass throughout the trajectory.